nt.number theory – Strange lacunary Lambert series related to the Liouville function

Although I have my own interest for the Liouville function, I will suppress it here as the question seems to be interesting in its own right.

It occurred to me when I saw an answer by GH from MO to Normal numbers, Liouville function, and the Riemann Hypothesis. That answer mentions the paper by Borwein and Coons where it is proved (among other things) that the functions $f_lambda(z)=sum_{ngeqslant1}lambda(n)z^n$ and $f_mu(z)=sum_{ngeqslant1}mu(n)z^n$ are both transcendental (here $lambda$ is the Liouville function and $mu$ is the Möbius function; just in case, let me recall that $lambda(n)=(-1)^{Omega(n)}$ where $Omega(n)$ is the number of prime factors of $n$ counted with multiplicities, while $mu(n)=(-1)^{omega(n)}$ when $n$ is the product of $omega(n)$ distinct primes and zero otherwise).

Initially I became curious whether these series have the unit circle as the analyticity boundary and, if yes, what can be said about their radial limit values at roots of unity. One natural thing to look at in this respect are the Lambert series for these functions. Although this is not directly related to the question, it is still related, so let me just say without proof, that$$f_lambda(z)=sum_{ngeqslant1}frac{tildelambda(n)z^n}{1-z^n},qquad f_mu(z)=sum_{ngeqslant1}frac{tildemu(n)z^n}{1-z^n}$$where $tildelambda$ and $tildemu$ are multiplicative with, for $p$ a prime, $tildelambda(p^k)$ is $(-1)^ktimes2$ while $tildemu(p^k)$ is $-2$ for $k=1$, $1$ for $k=2$ and $0$ for $k>2$.

What happened next was that I thought about representing these functions as logarithmic derivatives of some functions with nice infinite product expansions, and then modified them slightly thinking about obtaining sort of nicer infinite products. Doing that I stumbled upon the following:begin{multline*}z(1+f_lambda(z))=z+z^2-z^3-z^4+z^5+…+lambda(n)z^{n+1}+…\=frac z{1-z}-frac{2z^3}{1-z^3}-frac{2z^4}{1-z^4}+frac{2z^{12}}{1-z^{12}}-frac{2z^{13}}{1-z^{13}}+…end{multline*}

Surprised by this strange “jump” from 4 to 12 I looked at the exponents in this Lambert series and found that there are several other jumps of this length (from $n$ to $n+8$), many shorter jumps, as well as at least one still longer jump, from 4450 to 4459. Note that there are no jumps at all for $tildelambda$. So my question is,

is there any explanation for these strange jumps? Are their lengths bounded?

Some considerations around it. Certainly there are lots of jumps for $tildemu$, since it is zero on any number divisible by a cube; but they are much shorter: no longer than $4$ up to $n=5000$. The analogous “shift” for $mu$, that is, the Lambert series for $z(1+f_mu(z))$ has slightly longer jumps but still, it seems, essentially smaller than the shift for $lambda$ — for example, up to $n=5000$ it does not have jumps longer than 6. Maybe all this changes for larger $n$, I don’t know.

Another thing: the Wikipedia page on Lambert series that I link to above contains some recent additions about some Factorization theorems that seem to exhibit new exciting links between Lambert series and partition functions. In principle these theorems provide explicit expressions between the Maclaurin and Lambert series in very general situations. However I don’t readily see how to use them to explain these strange jumps.

I found two related questions on MO: Ordinary Generating Function for Mobius where the answers indicate that most likely there are no radial limits at all for $f_mu$ (so maybe I will ask a separate question about the other functions that appear here), and Lambert series identity with an answer that might be useful here, maybe also related to those factorization theorems.